The Nonlinear Finite Element Method: \\ Finally, A Technique for Three-Dimensional Stellar Evolution, Accretion Disks, {\it etc.}
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**Session 40 -- Computational Astrophysics I**
*Display presentation, Wednesday, 1, 1994, 9:20-6:30*

## [40.01] The Nonlinear Finite Element Method: \\ Finally, A Technique for Three-Dimensional Stellar Evolution, Accretion Disks, {\it etc.}

*D. L. Meier (Caltech/JPL)*
By adapting the techniques of Finite Element Analysis, the
investigator has developed a very general scheme for solving
multi-dimensional, non-linear, time-dependent astrophysical boundary
value problems in complex geometries. Applications of these techniques
include, but are not limited to, the structure and evolution of
magnetized rotating stars and planets, interacting binary stars,
magnetized thin and thick accretion disks, and full four dimensional
solutions to Einstein's equations. Our fully-implicit simulation code
solves boundary value equations in up to four (4) dimensions with second
or third order accuracy in space and time. The only constraint on the
grid in our code is that it be *topologically*
(hyper-)rectangular.
Therefore, the grid *geometry*
can be rather arbitrary, making it
possible for grid points to lie on complex-shaped core-halo interfaces,
photospheres, *etc.*
All coordinate transformations between the grid
and physical spaces are determined and performed internally in the code.
These properties of the method allow radiative boundary conditions to be
applied on grid boundaries (even ones with evolving shape), just as they have
been in one-dimensional stellar models in the past. In addition, the
local grid spacing and geometry can be adapted to the (multi-dimensional)
solution in order to obtain higher accuracy in regions of steep gradients.
Any motion of this adaptive grid with respect to the local medium is
automatically taken into account by the method in advective derivatives.

The basic Finite Element Method will be discussed along with the
modifications necessary for solving astrophysical problems and the steps
necessary for adapting this technique to massively-parallel supercomputers,
such as the Intel Paragon at Caltech and the Cray T3D at JPL. Some solutions
of simple multi-dimensional boundary value problems with and without adaptive
gridding will be presented. Recent results on more complex astrophysical
problems also will be discussed, pending availability.

**Wednesday
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